量子场论中的有限格距动量算子
Finite-range Lattice Momentum Operators for Quantum Field Theory
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中文总结 AI 辅助
该研究提出Z变换框架，构造有限冲激响应型有限格距动量算子，解决量子场论中费米子加倍问题，θ=π附近无幽灵波包且仅平面波相干传播。
中文摘要 AI 辅助
我们提出了一种Z变换框架，用于分析和合成量子场论中的有限格距动量算子。在该表述中，平移不变的格算子被表示为单位圆内复变量z的函数，从而可利用数字信号处理和有理逼近理论的工具分析其谱性质。在此框架下，费米子加倍问题被重新解释为离散动量算子在单位圆上出现多余零点的现象——这是奈奎斯特采样定理意义下的混叠现象，而幽灵态抑制的条件则被表述为对算子传递函数零点结构的精确约束。研究证明，不存在任何有理函数能同时满足所有要求的条件，这促使我们开发了此处提出的有限冲激响应方法。这种重新表述自然地给出了一类有限格距动量算子，其通过求解频域中的最小二乘逼近问题构造而成。所得有限冲激响应（FIR）算子在整个布里渊区内近似连续介质导数，且幽灵态抑制是通过谱逼近的精度实现的，而非通过添加对称性破缺的Wilson项或无限程非局域的SLAC导数。数值研究证实，在θ=π附近，仅平面波能相干传播，且这些平面波的群速度远超光速，这进一步将它们与物理的低能激发区分开；在θ=π附近不存在幽灵波包解。
英文摘要
We propose a Z-transform framework for the analysis and synthesis of finite range lattice momentum operators in quantum field theory. In this formulation, translation-invariant lattice operators are represented as functions of the complex variable $z$ in the unit circle, allowing their spectral properties to be analyzed using tools from digital signal processing and rational approximation theory. Within this framework, the fermion doubling problem is reinterpreted as the appearance of unwanted zeros of the discrete momentum operator on the unit circle --- an aliasing phenomenon in the sense of the Nyquist sampling theorem --- and the conditions for ghost suppression are expressed as precise constraints on the zero structure of the operator's transfer function. It is proven that no rational function can satisfy all required conditions simultaneously, motivating the finite impulse response approach developed here. This reframing naturally suggests a class of finite-range momentum operators, constructed by solving a least-squares approximation problem in the frequency domain. The resulting finite impulse response (FIR) operator approximates the continuum derivative across the full Brillouin zone, with ghost suppression achieved through the accuracy of the spectral approximation rather than through the addition of a symmetry-breaking Wilson term or the infinite-range nonlocal SLAC derivative. Numerical investigation confirms that near $θ= π$ only plane waves propagate coherently, and these exhibit group velocities far exceeding the speed of light, further distinguishing them from physical low-energy excitations. No ghost wave packet solutions exist near $θ= π$.
